Economic and financial modeling and planning is commonly used to estimate or predict the performance and outcome of real systems, given specific proposed sets of input data. An economic-based system will have many variables and influences which determine its behavior. A model is a mathematical expression or representation which predicts the outcome or behavior of the system under a variety of conditions. In one sense, it is relatively easy, in the past tense, to review historical data, understand its past performance, and state with relative certainty that the system's past behavior was influenced by the historical data. A much more difficult task, but one that is extremely valuable, is to generate a mathematical model of the system which predicts how the system will behave with a different set of data and assumptions. While foreseeing different outcomes with different sets of input data is inherently imprecise, i.e., no model can achieve 100% certainty, the field of probability and statistics has provided many tools which allow such predictions to be made with reasonable certainty and acceptable levels of confidence.
In its basic form, the economic model can be viewed as a predicted or anticipated outcome, as driven by a given set of input data and assumptions. The input data is processed through a mathematical expression representing either the expected or current behavior of the real system. The mathematical expression is formulated or derived from principals of probability and statistics, often by analyzing historical data and corresponding known outcomes, to achieve a best fit of the expected behavior of the system to other sets of data. In other words, the model should be able to predict the outcome or response of the system to a specific set of data being considered or proposed, within a given level of confidence, or an acceptable level of uncertainty. As a simple test of the quality of the model, if historical data is processed through the model and the outcome of the model, using the historical data, is closely aligned with the known historical outcome, then the model is considered to have a high confidence level over the interval. The model should then do a good job of predicting outcomes of the system to different sets of input data.
Most, if not all, modeling in statistics and economics is based on an assumption, that the underlying process is stationary and doesn't change with time. Unfortunately, most processes of commercial interest do change with time, i.e., non-stationary. The time dependency poses a challenge for researchers in statistics and economics.
Practitioners often times use ad-hoc methods to address the problem of non-stationary processes. For example, in the retail industry it is common to develop a category seasonality index from historical sales. The index is then used to “de-seasonalize” the sales. The category seasonality index approach is not a rigorous method and runs into many problems. There are two major drawbacks to this method: 1) often times products have unique demand profiles that are not in sync with the overall categories demand, and 2) causal factors like promotional and pricing activity can be absorbed into the seasonality index. Both of these factors can lead to biased parameter estimates. It is not uncommon for this method to lead to price elasticities with the wrong sign.
Consider the application of economic modeling to retail merchandising. In most segments of the industry, there is an inverse relationship between the price of goods or services and volume or unit sales. As a general proposition, as price increases, unit sales decrease, and as price decreases, unit sales increase. However, ascertaining the relative magnitudes of the opposing movements, as well as evaluating various other factors influencing the basic parameters of price and sales, often requires sophisticated modeling tools.
Most commercially-useful economic models use highly complex algorithms and take into account many different factors and parameters to compute the predicted outcome. One such economic model is described in Kungl. Vetenskapsakademien, The Royal Swedish Academy of Sciences, Stockholm, Sweden, entitled “Time-Series Econometrics: Cointegration and Autoregressive Conditional Heteroskedasticity” by Robert Engle and Clive Granger, which describes macroeconomic and financial economic models in terms of nonstationarity and time-varying volatility over a time series. Engle and Granger received the 2003 Nobel Prize for their work on modeling non-stationary processes. Granger is credited with co-integration which is a method for modeling the difference of two time series that have the same underlying non-stationary process. If the model is linear then the non-stationary process cancels out and the system can be accurately modeled. Engle is credited with the autoregressive conditional heteroskedasticity (ARCH) process which is a method for accounting for the variance or volatility of the time series.
While these methods have provided a tool for modeling macro-economic systems they are not very extensible to micro-economic systems. In micro-economic systems, the assumption of a Gaussian distribution breaks down and more complicated “non-linear” likelihood functions must be deployed, which is a problem for both methods listed above.
For purposes of the present illustration, a simplified model of price and unit sales is considered. FIG. 1 illustrates basic time series plots of price versus time and unit sales versus time. The time series plots illustrate a particular category or grouping of related goods or services. In FIG. 1, from time t0 to time t1, the price is $12 as shown by trace 12. From time t0 to time t1, the predicted unit sales, as shown by trace 14, is steady at 1000 units. At time t1, trace 12 (price), drops from $12 to $9. As a result of market forces associated with the price reduction, the model predicts that trace 14 (unit sales) will rise from 1000 to 2000 units. At time t2, trace 12 (price) goes back to $12 and trace 14 (unit sales) returns to 1000 units. The simplified model receives price as input data and predicts how the unit sales will follow, i.e., how unit sales are affected by different prices, or what pricing levels will achieve desired goals of unit sales.
The standard model uses parametric measures having a known or given statistical distribution for the category as a whole. The prediction algorithm may be as simple as an average of sales across each time period, or it may use more complex functions. For example, the model of unit sales (US) as function of price (P), may be given as expressed in equation (1):US(P)=Ae−βp  (1)
The base demand parameters of the above Gaussian distribution are A and β and the parametric model assumes that unit sales or demand follows that distribution as a function of price across the entire time series.
It is common practice for standard parametric modeling techniques to first de-seasonalize the sales data at a category level. The category level is a grouping of related products, e.g., 1000 different products in the staple foods, or health and beauty aids, or home improvement products. The data is filtered or adjusted to remove such factors as seasonal variation, e.g., holiday, summer, winter sales, across the category as a whole. Once de-seasonalized, the standard parametric model assumes that the demand is flat, i.e., that only price and promotion (merchandising and marketing) remain to affect demand. A flat demand presumes that there are no other material factors influencing demand.
However, in real economic systems, such as retail merchandising, the presumption of a flat demand, even after de-seasonalizing the category data, is often invalid or inaccurate and can lead to biased parameter estimates. In many cases, variation in demand of specific products within the category over time will influence the category-level demand response. Yet, in parametric modeling, demand-related factors are not modeled at the product level and, as such, do not take into account such time-varying factors. Even though the demand has been adjusted for seasonality at the category level, other time dependent factors which occur at the product level may still affect demand. These time dependent product-level factors, which are not taken into account in standard parametric modeling, include factors such as product introduction or product discontinuation.
The introduction of new products and/or discontinuation of existing products can have material impact on category-level demand, either in the positive or negative direction. If the new product is hot or en vogue from some event or need, demand may be sky high in its own right, separate and apart from the price or promotional efforts. If the new product is unknown or yet to be accepted, demand may need to ramp up over time, especially if the new product faces stiff competition. Product discontinuation may cause consumers to shy away if the product is perceived as out-dated, or consumers may stock-pile the discontinued product for fear of no suitable replacement.
Another time dependent product-level factor involves products within a category that have different seasonal response than the category as a whole. If the seasonal demand for a given category of products goes down, but one or more products within that category experiences an increase in demand during the same season, or vice versa, then the parametric model goes astray. For example, in the fall, the category of fresh produce in retail food stores may decline because of the end of the growing season. However, come the end of October, sales of pumpkins and apples have the opposite seasonal demand response, i.e., Halloween is prime-time for pumpkins.
Since parametric modeling does not account for such time dependent product-level factors, which may have a material impact on category-level demand, the resulting parametric model loses its ability to model outcomes with high confidence. The parametric-derived demand profile curve does not fit the historical data and does not necessarily predict demand profiles and parameters with reasonable certainty. The measure of price elasticity, which indicates how unit sales respond to price change, becomes distorted, i.e., price elasticity may be too high or too low, and may even register as negative in certain scenarios, which makes no sense. While it is possible to manually adjust or tune parametric models to compensate for such time dependent product-level factors which influence demand, the tuning exercise is time consuming and impractical in many situations. Most users of economic models do not favor any model that requires continuous tweaking to get reasonable predictions. The manual tuning effort multiplies when one considers the many different products, locations, and venues which may be involved.
A need exists to accurately model time dependent parameters within a likelihood function.